重尾扰动下近非平稳自回归模型的Bootstrap推断

考虑近非平稳一阶自回归模型,Xt=θXt-1+ut,其中θn=1-γ/n,γ为一固定常数,{ut}为一重尾随机扰动项.对θn及其最小二乘估计(θ)n,采用bootstrap再抽样方法逼近(θ)n-θn的分布,并证明了该再抽样方法的渐近有效性.     

On Monte Carlo methods for Bayesian multivariate regression models with heavy-tailed errors

We consider Bayesian analysis of data from multivariate linear regression models whose errors have a distribution that is a scale mixture of normals. Such models are used to analyze data on financial returns, which are notoriously heavy-tailed. Let π denote the intractable posterior density that results when this regression model is combined with the standard non-informative prior on the unknown regression coefficients and scale matrix of the errors. Roughly speaking, the posterior is proper if and only if n≥d+k, where n is the sample size, d is the dimension of the response, and k is number of covariates. We provide a method of making exact draws from π in the special case where n=d+k, and we study Markov chain Monte Carlo (MCMC) algorithms that can be used to explore π when n>d+k. In particular, we show how the Haar PX-DA technology studied in Hobert and Marchev (2008) [11] can be used to improve upon Liu’s (1996) [7] data augmentation (DA) algorithm. Indeed, the new algorithm that we introduce is theoretically superior to the DA algorithm, yet equivalent to DA in terms of computational complexity. Moreover, we analyze the convergence rates of these MCMC algorithms in the important special case where the regression errors have a Student’s t distribution. We prove that, under conditions on n, d, k, and the degrees of freedom of the t distribution, both algorithms converge at a geometric rate. These convergence rate results are important from a practical standpoint because geometric ergodicity guarantees the existence of central limit theorems which are essential for the calculation of valid asymptotic standard errors for MCMC based estimates.     

Asymptotics of R-, MD- and LAD-estimators in linear regression models with long range dependent errors

Summary This paper establishes the uniform closeness of a weighted residual empirical process to its natural estimate in the linear regression setting when the errors are Gaussian, or a function of Gaussian random variables, that are strictly stationary and long range dependent. This result is used to yield the asymptotic uniform linearity of a class of rank statistics in linear regression models with long range dependent errors. The latter result, in turn, yields the asymptotic distribution of the Jaeckel (1972) rank estimators. The paper also studies the least absolute deviation and a class of certain minimum distance estimators of regression parameters and the kernel type density estimators of the marginal error density when the errors are long range dependent.Research of this author was partly supported by the NSF grant: DMS-9102041     

Robust semiparametric M-estimation and the weighted bootstrap

M-estimation is a widely used technique for statistical inference. In this paper, we study properties of ordinary and weighted M-estimators for semiparametric models, especially when there exist parameters that cannot be estimated at the convergence rate. Results on consistency, rates of convergence for all parameters, and consistency and asymptotic normality for the Euclidean parameters are provided. These results, together with a generic paradigm for studying semiparametric M-estimators, provide a valuable extension to previous related research on semiparametric maximum-likelihood estimators (MLEs). Although penalized M-estimation does not in general fit in the framework we discuss here, it is shown for a great variety of models that many of the forgoing results still hold, including the consistency and asymptotic normality of the Euclidean parameters. For semiparametric M-estimators that are not likelihood based, general inference procedures for the Euclidean parameters have not previously been developed. We demonstrate that our paradigm leads naturally to verification of the validity of the weighted bootstrap in this setting. For illustration, several examples are investigated in detail. The new M-estimation framework and accompanying weighted bootstrap technique shed light on a universal way of investigating semiparametric models.     

On linear models with long memory and heavy-tailed errors

We consider the robust estimation of regression parameters in linear models with long memory and heavy-tailed errors. Asymptotic Bahadur-type representations of robust estimates are developed and their limiting distributions are obtained. It is shown that the limiting distributions are very different from those obtained under short memory. A simulation study is carried out to compare the performance of various asymptotic representations.     

Asymptotics for the Euclidean TSP with power weighted edges

Summary LetU ₁,...,U_n denote i.i.d. random variables with the uniform distribution on [0, 1]², and letT ²T²(U₁,...,U_n) denote the shortest tour throughU ₁,...,U_n with square-weighted edges. By drawing on the quasi-additive structure ofT ² and the boundary rooted dual process, it is shown that lim _n E T ²(U ₁,...,U_n)= for some finite constant .This work was supported in part by NSF Grant DMS-9200656, Swiss National Foundation Grant 21-298333.90, and the US Army Research Office through the Mathematical Sciences Institute of Cornell University, whose assistance is gratefully acknowledged     

Asymptotic Bayesian structure learning using graph supports for Gaussian graphical models

The theory of Gaussian graphical models is a powerful tool for independence analysis between continuous variables. In this framework, various methods have been conceived to infer independence relations from data samples. However, most of them result in stepwise, deterministic, descent algorithms that are inadequate for solving this issue. More recent developments have focused on stochastic procedures, yet they all base their research on strong a priori knowledge and are unable to perform model selection among the set of all possible models. Moreover, convergence of the corresponding algorithms is slow, precluding applications on a large scale. In this paper, we propose a novel Bayesian strategy to deal with structure learning. Relating graphs to their supports, we convert the problem of model selection into that of parameter estimation. Use of non-informative priors and asymptotic results yield a posterior probability for independence graph supports in closed form. Gibbs sampling is then applied to approximate the full joint posterior density. We finally give three examples of structure learning, one from synthetic data, and the two others from real data.     

Strong consistency of least-squares estimation in linear regression models with vague concepts

Linear regression models with vague concepts extend the classical single equation linear regression models by admitting observations in form of fuzzy subsets instead of real numbers. They have recently been introduced [cf. Krätschmer, Induktive statistik auf basis unscharfer meßkonzepte am beispiel linearer regressionsmodelle, Unpublished Habilitation Monograph, Faculty of Law and Economics of the University of Saarland, Saarbrücken, 2001] to improve the empirical meaningfulness of the relationship between the involved items by a more sensitive attention to the problems of data measurement, in particular the fundamental problem of adequacy. The parameters of such models are still real numbers, and a method of estimation can be applied which extends directly the ordinary least-squares method. This paper deals with some first asymptotic properties of estimators obtained by the method. Firstly, strong consistency will be shown, and secondly, the convergence rate will be investigated. The later result will be the starting point for a future study which will calculate the limit distributions of the estimators.     

Partial sum process to check regression models with multiple correlated response: With an application for testing a change-point in profile data

We consider regression models with multiple correlated responses for each design point. Under the null hypothesis, a linear regression is assumed. For the least-squares residuals of this linear regression, we establish the limit of the partial sums. This limit is a projection on a certain subspace of the reproducing Kernel Hilbert space of a multivariate Brownian motion. Based on this limit, we propose a significance test of Kolmogorov-Smirnov type to test the null hypothesis and show that this result can be used to study a change-point problem in the case of linear profile data (panel data). We compare our proposed method, which does not rely on any distributional assumptions, with the likelihood ratio test in a simulation study.     

Likelihood ratio tests for continuous monotone hazards with an unknown change point

We consider the likelihood ratio tests (LRT) for two continuous monotone hazards with an unknown change point. We establish the convergence in distribution and weak convergence of LRT. Simulation studies show that the proposed tests compare favorably to other existing tests.     

THE RANDOM WEIGHTING APPROXIMATION FOR THE VIRTUAL SYSTEM METHOD

In this paper, the random weighting method is applied to the exponential system, and we construct the approximation to the estimates of the system parameters with an accuracy of o(n^-1/2) as n→∞.     

Poisson approximation for the number of large digits of inhomogeneousf-expansions

We determine the exact rate of Poisson approximation and give a second-order Poisson-Charlier expansion for the number of excedances of a given levelL _n among the firstn digits of inhomogeneousf-expansions of real numbers in the unit interval. The application of this general result to homogeneousf-expansions and, in particular, to regular continued fraction expansions provides not only a generalization but also a strengthening of a classical Poisson limit theorem due to W. Doeblin.     

Limit distributions of least squares estimators in linear regression models with vague concepts

Linear regression models with vague concepts extend the classical single equation linear regression models by admitting observations in form of fuzzy subsets instead of real numbers. They have lately been introduced (cf. [V. Krätschmer, Induktive Statistik auf Basis unscharfer Meßkonzepte am Beispiel linearer Regressionsmodelle, unpublished postdoctoral thesis, Faculty of Law and Economics of the University of Saarland, Saarbrücken, 2001; V. Krätschmer, Least squares estimation in linear regression models with vague concepts, Fuzzy Sets and Systems, accepted for publication]) to improve the empirical meaningfulness of the relationships between the involved items by a more sensitive attention to the problems of data measurement, in particular, the fundamental problem of adequacy. The parameters of such models are still real numbers, and a method of estimation can be applied which extends directly the ordinary least squares method. In another recent contribution (cf. [V. Krätschmer, Strong consistency of least squares estimation in linear regression models with vague concepts, J. Multivar. Anal., accepted for publication]) strong consistency and -consistency of this generalized least squares estimation have been shown. The aim of the paper is to complete these results by an investigation of the limit distributions of the estimators. It turns out that the classical results can be transferred, in some cases even asymptotic normality holds.     

Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

We discuss joint temporal and contemporaneous aggregation of N$N$ independent copies of AR(1) process with random-coefficient a∈[0,1)$a\in [0,1)$ when N$N$ and time scale n$n$ increase at different rate. Assuming that a$a$ has a density, regularly varying at a=1$a=1$ with exponent −1<β<1$-1<\beta <1$, different joint limits of normalized aggregated partial sums are shown to exist when N^1/(1+β)/n$N^{1/(1+\beta )}/n$ tends to (i) ∞$\infty$, (ii) 0$0$, (iii) 0<μ<∞$0<\mu <\infty$. The limit process arising under (iii) admits a Poisson integral representation on (0,∞)×C(R)$(0,\infty )\times C\left(\mathbb{R}\right)$ and enjoys ‘intermediate’ properties between fractional Brownian motion limit in (i) and sub-Gaussian limit in (ii).     

Empirical likelihood confidence region for parameter in the errors-in-variables models

This paper proposes a constrained empirical likelihood confidence region for a parameter β₀ in the linear errors-in-variables model: Y_i=x_i^τβ₀+ε_i,X_i=x_i+u_i,(1?i?n), which is constructed by combining the score function corresponding to the squared orthogonal distance with a constrained region of β₀. It is shown that the coverage error of the confidence region is of order n⁻¹, and Bartlett corrections can reduce the coverage errors to n⁻². An empirical Bartlett correction is given for practical implementation. Simulations show that the proposed confidence region has satisfactory coverage not only for large samples, but also for small to medium samples.     

Experimental design for estimating the optimum point in a response surface

The problem of optimal experimental design for response optimization is considered. The optimal point (control)x ^* of a response surface is to be determined by estimating the response parameters from measurements performed at design pointsx ⁱ,i=1,...,N. Classical sequential approaches for choosing thex ⁱ's are recalled. A loss function related to the issue of response optimization is used to define control-oriented design criteria. The design policies differ depending on whether least-squares or minimum risk estimation is used to estimate. Connections between various criteria suggested in the literature are exhibited. Special attention is given to quadratic model responses. Most approaches presented assume that the response is correctly described by a given parametric function over the region of interest. Possible deterministic departures from this function raise the problem of model robustness, and the literature on the subject is briefly surveyed.